Let G be a graph of diameter at least three. Can you find an upper bound on the diameter of the complement of G? Prove your findings!

Let G be a connected graph and sq(G) be a graph which contains all vertices and edges of G and moreover edges joining every pair of vertices that were in G at distance 2. In other words, xy is an edge of sq(G) if and only if the distance of x and y in G is at most 2. Prove that sq(G) is 2-connected.

... to n(n +1)/2 > 1, then w must have at least one edge connecting some node ... of G, so the path v ~ u -> w is in G and connects v and w. Thus G is connected. ...

Hamiltonian Graphs and 2-Connected Graphs. ... Hamiltonian Graphs and 2-Connected Graphs are investigated. The solution is detailed and well presented. ...

... components), there is always a path connecting them, in ... if one is disconnected, then the other is connected. An elementary theorem in graph theory is applied ...

... The edges that connect the vertices can be lines ... Then, these vertices can be connected using edges to ... the vertices and the weighted edges connecting the cities ...

... that are connected to v in G via an edge. By the definition of graph isomorphism, we know that, for every vertex v1 in V(G), there is an edge connecting v and ...

... if and only if there is no edge that connects v1 and ... if and only if there is an edge connecting f(v1 ... There certainly are graphs on n vertices with n ≡ 0 (mod 4 ...

... distance of 6, and an edge connecting it with ... every computer has to be directly connected to the ... a minimal spanning tree approach to connect the centralized ...

... the graph D is connected, there is a shortest path from 'e' the the vertex set of T in the undirected base graph. Let 'f' be the edge that connects this path ...